The shape of most animal cells is controlled by the actin cortex, a thin, isotropic network of dynamic actin filaments (F-actin) situated just beneath the plasma membrane. The cortex is held far from equilibrium by both active stresses and turnover: Myosin-II molecular motors drive deformations required for cell division, migration, and tissue morphogenesis, while turnover of the molecular components of the actin cortex relax stress and facilitate network reorganization. While many aspects of F-actin network viscoelasticity are well-characterized in the presence and absence of motor activity, a mechanistic understanding of how non-equilibrium actin turnover contributes to stress relaxation is still lacking. To address this, we developed a reconstituted in vitro system wherein the steady-state length and turnover rate of F-actin in entangled solutions are controlled by the actin regulatory proteins cofilin, profilin, and formin, which sever, recycle, and nucleate filaments, respectively. Cofilin-mediated severing accelerates the turnover and spatial reorganization of F-actin, without significant changes to filament length. Microrheology measurements demonstrate that cofilin-mediated severing is a single-timescale mode of stress relaxation that tunes the low-frequency viscosity over two orders of magnitude. These findings serve as the foundation for understanding the mechanics of more physiological F-actin networks with turnover, and inform an updated microscopic model of single-filament turnover. They also demonstrate that polymer activity, in the form of ATP hydrolysis on F-actin coupled to nucleotide-dependent cofilin binding, is sufficient to generate a form of active matter wherein asymmetric filament disassembly preserves filament number in spite of sustained severing.